Putting aside the idea that "analog" doesn't have overflow. (Any physical implementation of that analog signal would have limits in the real world. If nothing else, then how much current or voltage your PSU can supply, or how much current can go through your wire.)

But that is not even the real issue. The real issue that it is not analog, it is just "analog" with quotes. The neural network is executed by a digital computer, on digital hardware. So that "analog" is just the underlying float type of the of the CPU or GPU executing the network. That is still very much digital. Just on a different abstraction level.

Of course it is, there is whole business around overflowing analog signals in right way in analog guitar effects pedal